Different ways of computing a limit (Part 4)

One of my colleagues placed the following problem on an exam for his Calculus II course…

and was impressed by the variety of correct responses that he received. I thought it would be fun to discuss some of the different ways that this limit can be computed.

Method #4. The geometric approach. The numbers and can be viewed as two sides of a right triangle with legs and and hypotenuse . So as gets larger and larger, the longer leg will get closer and closer in length to the length of the hypotenuse. Therefore, the ratio of the length of the hypotenuse to the length of the longer leg must be 1.